I do not know, never rendered fat myself.
However; when I think Oil, I think 'Thin and Runny, when I think Grease, I think 'Thick' and...well 'greasy' - like anything else. Think 'Automotive'; you dont put oil on bearings and you dont slap gobs of grease down the pipe labeled 'Grease'.
I read an article, somewhere, describing the process of rendering and it described how the fat would break down with a lighter area that would rise to the top, it said this would be filtered off and used for "medical, makeup, etc" (was not a ML article) and heavier stuff would sink to bottom, this would be the lard...then it said something (??) about the in between layer(s).
Perhaps if one rendered longer the 'oils' on top would burn off?? But I don't think you would want to burn off all the oils for I feel that would leave....perhaps...less desirable 'slick' "grease"??
I imagine like anything else there is an art and science to it all.
I know one on YouTube who renders Raccoon and even Skunk fat but he uses it for 'Candles'....made me wonder about it as a lube - he got a decent amount out of his **** for the video.